This can be effected without evidence or trial or a right to an appeal in front of an objective party.
Not so. If a person is charged with a crime for violating such a code, (or refusing to leave when ordered under such a code) they could defend on the grounds that it is unreasonable, unauthorized, or violates that person's constitutional or statutory rights. Or, if a person has been ordered to leave, the person could comply and then seek an injunction forbidding future enforcement of the regulation. Such methods have been used in the past to challenge the lawfulness of administrative regulations.
The Colorado code CRS 24-90-109. Powers and duties of board of trustees says that:
(1) The board of trustees shall:
(a) Adopt such bylaws, rules, and regulations for its own guidance and policies for the governance of the library as it deems expedient. ...
(b) Have custody of all property of the library, including rooms or buildings constructed, leased, or set apart therefor;
(c) Employ a director and, upon the director's recommendation, employ such other employees as may be necessary. The duties of the director shall include, but not limited to:
(I) Implementing the policies adopted by the board of trustees pursuant to paragraph (a) of subsection (1) of this section;
(III) Performing all other acts necessary for the orderly and efficient management and control of the library.
This law seems to authorize libraries to adopt and enforce codes of conduct for persons using the library.
This page from the CO State library development agency gives example policies that libraries are encouraged to model their policies on. In the section under "Library Use/Behavior" there are three example policies. All of these list various prohibited acts, which seem relatively reasonable to me. Two of the three include an explicit appeals process for serious violations.
You do not indicate what sort of behavior you have encaged in for which the library may wish to ban you.
In general, government facilities are allowed to make reasonable regulations for members of the public using those facilities, and it is not a violation of people's Constitutional rights to make and enforce such regulations. However, that depends highly on what the regulations are. A regulation limiting access by race would obviously be struck down. A regulation prohibiting shouting, even though it impacts speech, would be permitted as a content-neutral regulation of "time, place and manner". In short it would depend very much on the specific regulation, and what rights it is alleged to violate.